Gaurish, host of, For the love of Mathematics, gives me another topic for today’s A To Z entry. I think the subject got away from me. But I also like where it got.
Something about ‘5’ that you only notice when you’re a kid first learning about numbers. You know that it’s a prime number because it’s equal to 1 times 5 and nothing else. You also know that once you introduce fractions, it’s equal to all kinds of things. It’s 10 times one-half and it’s 15 times one-third and it’s 2.5 times 2 and many other things. Why, you might ask the teacher, is it a prime number if it’s got a million billion trillion different factors? And when every other whole number has as many factors? If you get to the real numbers it’s even worse yet, although when you’re a kid you probably don’t realize that. If you ask, the teacher probably answers that it’s only the whole numbers that count for saying whether something is prime or not. And, like, 2.5 can’t be considered anything, prime or composite. This satisfies the immediate question. It doesn’t quite get at the underlying one, though. Why do integers have prime numbers while real numbers don’t?
To maybe have a prime number we need a ring. This is a creature of group theory, or what we call “algebra” once we get to college. A ring consists of a set of elements, and a rule for adding them together, and a rule for multiplying them together. And I want this ring to have a multiplicative identity. That’s some number which works like ‘1’: take something, multiply it by that, and you get that something back again. Also, I want this multiplication rule to commute. That is, the order of multiplication doesn’t affect what the result is. (If the order matters then everything gets too complicated to deal with.) Let me say the things in the set are numbers. It turns out (spoiler!) they don’t have to be. But that’s how we start out.
Whether the numbers in a ring are prime or not depends on the multiplication rule. Let’s take a candidate number that I’ll call ‘a’ to make my writing easier. If the only numbers whose product is ‘a’ are the pair of ‘a’ and the multiplicative identity, then ‘a’ is prime. If there’s some other pair of numbers that give you ‘a’, then ‘a’ is not prime.
The integers — the positive and negative whole numbers, including zero — are a ring. And they have prime numbers just like you’d expect, if we figure out some rule about how to deal with the number ‘-1’. There are many other rings. There’s a whole family of rings, in fact, so commonly used that they have shorthand. Mathematicians write them as “Zn”, where ‘n’ is some whole number. They’re the integers, modulo ‘n’. That is, they’re the whole numbers from ‘0’ up to the number ‘n-1’, whatever that is. Addition and multiplication work as they do with normal arithmetic, except that if the result is less than ‘0’ we add ‘n’ to it. If the result is more than ‘n-1’ we subtract ‘n’ from it. We repeat that until the result is something from ‘0’ to ‘n-1’, inclusive.
(We use the letter ‘Z’ because it’s from the German word for numbers, and a lot of foundational work was done by German-speaking mathematicians. Alternatively, we might write this set as “In”, where “I” stands for integers. If that doesn’t satisfy, we might write this set as “Jn”, where “J” stands for integers. This is because it’s only very recently that we’ve come to see “I” and “J” as different letters rather than different ways to write the same letter.)
These modulo arithmetics are legitimate ones, good reliable rings. They make us realize how strange prime numbers are, though. Consider the set Z4, where the only numbers are 0, 1, 2, and 3. 0 times anything is 0. 1 times anything is whatever you started with. 2 times 1 is 2. Obvious. 2 times 2 is … 0. All right. 2 times 3 is 2 again. 3 times 1 is 3. 3 times 2 is 2. 3 times 3 is 1. … So that’s a little weird. The only product that gives us 3 is 3 times 1. So 3’s a prime number here. 2 isn’t a prime number: 2 times 3 is 2. For that matter even 1 is a composite number, an unsettling consequence.
Or then Z5, where the only numbers are 0, 1, 2, 3, and 4. Here, there are no prime numbers. Each number is the product of at least one pair of other numbers. In Z6 we start to have prime numbers again. But Z7? Z8? I recommend these questions to a night when your mind is too busy to let you fall asleep.
Prime numbers depend on context. In the crowded universe of all the rational numbers, or all the real numbers, nothing is prime. In the more austere world of the Gaussian Integers, familiar friends like ‘3’ are prime again, although ‘5’ no longer is. We recognize that as the product of and , themselves now prime numbers.
So given that these things do depend on context. Should we care? Or let me put it another way. Suppose we contact a wholly separate culture, one that we can’t have influenced and one not influenced by us. It’s plausible that they should have a mathematics. Would they notice prime numbers as something worth study? Or would they notice them the way we notice, say, pentagonal numbers, a thing that allows for some pretty patterns and that’s about it?
Well, anything could happen, of course. I’m inclined to think that prime numbers would be noticed, though. They seem to follow naturally from pondering arithmetic. And if one has thought of rings, then prime numbers seem to stand out. The way that Zn behaves changes in important ways if ‘n’ is a prime number. Most notably, if ‘n’ is prime (among the whole numbers), then we can define something that works like division on Zn. If ‘n’ isn’t prime (again), we can’t. This stands out. There are a host of other intriguing results that all seem to depend on whether ‘n’ is a prime number among the whole numbers. It seems hard to believe someone could think of the whole numbers and not notice the prime numbers among them.
And they do stand out, as these reliably peculiar things. Many things about them (in the whole numbers) are easy to prove. That there are infinitely many, for example, you can prove to a child. And there are many things we have no idea how to prove. That there are infinitely many primes which are exactly two more than another prime, for example. Any child can understand the question. The one who can prove it will win what fame mathematicians enjoy. If it can be proved.
They turn up in strange, surprising places. Just in the whole numbers we find some patches where there are many prime numbers in a row (Forty percent of the numbers 1 through 10!). We can find deserts; we know of a stretch of 1,113,106 numbers in a row without a single prime among them. We know it’s possible to find prime deserts as vast as we want. Say you want a gap between primes of at least size N. Then look at the numbers (N+1)! + 2, (N+1)! + 3, (N+1)! + 4, and so on, up to (N+1)! + N+1. None of those can be prime numbers. You must have a gap at least the size N. It may be larger; how we know that (N+1)! + 1 is a prime number?
No telling. Well, we can check. See if any prime number divides into (N+1)! + 1. This takes a long time to do if N is all that big. There’s no formulas we know that will make this easy or quick.
We don’t call it a “prime number” if it’s in a ring that isn’t enough like the numbers. Fair enough. We shift the name to “prime element”. “Element” is a good generic name for a thing whose identity we don’t mean to pin down too closely. I’ve talked about the Gaussian Primes already, in an earlier essay and earlier in this essay. We can make a ring out of the polynomials whose coefficients are all integers. In that, is a prime. So is . If this hasn’t given you some ideas what other polynomials might be primes, then you have something else to ponder while trying to sleep. Thinking of all the prime polynomials is likely harder than you can do, though.
Prime numbers seem to stand out, obvious and important. Humans have known about prime numbers for as long as we’ve known about multiplication. And yet there is something obscure about them. If there are cultures completely independent of our own, do they have insights which make prime numbers not such occult figures? How different would the world be if we knew all the things we now wonder about primes?